LMFDB - Embedded newform 6936.2.a.bl.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6936,2,Mod(1,6936)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6936, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6936.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6936 = 2^{3} \cdot 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6936.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$55.3842388420$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.77984694272.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 10x^{6} + 36x^{5} + 39x^{4} - 96x^{3} - 56x^{2} + 64x - 8 $ x^8 - 4x^7 - 10x^6 + 36x^5 + 39x^4 - 96x^3 - 56x^2 + 64*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 408)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.149074$ of defining polynomial
Character	$\chi$	$=$	6936.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.11290 q^{5} -1.30012 q^{7} +1.00000 q^{9} -1.03004 q^{11} -6.68521 q^{13} -3.11290 q^{15} -0.0523478 q^{19} +1.30012 q^{21} +2.44426 q^{23} +4.69014 q^{25} -1.00000 q^{27} -1.16182 q^{29} +7.49914 q^{31} +1.03004 q^{33} -4.04713 q^{35} -4.26691 q^{37} +6.68521 q^{39} -2.59407 q^{41} +11.6733 q^{43} +3.11290 q^{45} +7.75431 q^{47} -5.30969 q^{49} -8.63508 q^{53} -3.20642 q^{55} +0.0523478 q^{57} +7.72763 q^{59} +6.99620 q^{61} -1.30012 q^{63} -20.8104 q^{65} +0.660863 q^{67} -2.44426 q^{69} -7.45066 q^{71} +1.38604 q^{73} -4.69014 q^{75} +1.33918 q^{77} -16.6708 q^{79} +1.00000 q^{81} +7.05834 q^{83} +1.16182 q^{87} -18.5036 q^{89} +8.69155 q^{91} -7.49914 q^{93} -0.162953 q^{95} -7.51406 q^{97} -1.03004 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} - 4 q^{5} + 8 q^{9} - 4 q^{11} - 4 q^{13} + 4 q^{15} + 4 q^{19} + 4 q^{23} - 4 q^{25} - 8 q^{27} - 8 q^{31} + 4 q^{33} + 8 q^{35} - 16 q^{37} + 4 q^{39} - 20 q^{41} + 28 q^{43} - 4 q^{45}+ \cdots - 4 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 - 4 * q^5 + 8 * q^9 - 4 * q^11 - 4 * q^13 + 4 * q^15 + 4 * q^19 + 4 * q^23 - 4 * q^25 - 8 * q^27 - 8 * q^31 + 4 * q^33 + 8 * q^35 - 16 * q^37 + 4 * q^39 - 20 * q^41 + 28 * q^43 - 4 * q^45 + 24 * q^47 - 8 * q^49 + 8 * q^53 - 4 * q^55 - 4 * q^57 + 16 * q^59 - 24 * q^61 - 28 * q^65 - 4 * q^69 - 40 * q^73 + 4 * q^75 + 16 * q^77 - 8 * q^79 + 8 * q^81 + 32 * q^83 + 16 * q^89 + 8 * q^93 - 4 * q^95 - 48 * q^97 - 4 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.11290	1.39213	0.696065	−	0.717978i	$-0.254932\pi$
$5$	3.11290	1.39213	0.696065	+	0.717978i	$0.254932\pi$
$6$	0	0
$7$	−1.30012	−0.491398	−0.245699	−	0.969346i	$-0.579018\pi$
$7$	−1.30012	−0.491398	−0.245699	+	0.969346i	$0.579018\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.03004	−0.310570	−0.155285	−	0.987870i	$-0.549630\pi$
$11$	−1.03004	−0.310570	−0.155285	+	0.987870i	$0.549630\pi$
$12$	0	0
$13$	−6.68521	−1.85414	−0.927071	−	0.374885i	$-0.877682\pi$
$13$	−6.68521	−1.85414	−0.927071	+	0.374885i	$0.877682\pi$
$14$	0	0
$15$	−3.11290	−0.803747
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−0.0523478	−0.0120094	−0.00600470	−	0.999982i	$-0.501911\pi$
$19$	−0.0523478	−0.0120094	−0.00600470	+	0.999982i	$0.501911\pi$
$20$	0	0
$21$	1.30012	0.283709
$22$	0	0
$23$	2.44426	0.509663	0.254832	−	0.966985i	$-0.417980\pi$
$23$	2.44426	0.509663	0.254832	+	0.966985i	$0.417980\pi$
$24$	0	0
$25$	4.69014	0.938028
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.16182	−0.215744	−0.107872	−	0.994165i	$-0.534404\pi$
$29$	−1.16182	−0.215744	−0.107872	+	0.994165i	$0.534404\pi$
$30$	0	0
$31$	7.49914	1.34689	0.673443	−	0.739239i	$-0.264815\pi$
$31$	7.49914	1.34689	0.673443	+	0.739239i	$0.264815\pi$
$32$	0	0
$33$	1.03004	0.179308
$34$	0	0
$35$	−4.04713	−0.684090
$36$	0	0
$37$	−4.26691	−0.701475	−0.350738	−	0.936474i	$-0.614069\pi$
$37$	−4.26691	−0.701475	−0.350738	+	0.936474i	$0.614069\pi$
$38$	0	0
$39$	6.68521	1.07049
$40$	0	0
$41$	−2.59407	−0.405126	−0.202563	−	0.979269i	$-0.564927\pi$
$41$	−2.59407	−0.405126	−0.202563	+	0.979269i	$0.564927\pi$
$42$	0	0
$43$	11.6733	1.78016	0.890080	−	0.455804i	$-0.150648\pi$
$43$	11.6733	1.78016	0.890080	+	0.455804i	$0.150648\pi$
$44$	0	0
$45$	3.11290	0.464044
$46$	0	0
$47$	7.75431	1.13108	0.565541	−	0.824720i	$-0.308667\pi$
$47$	7.75431	1.13108	0.565541	+	0.824720i	$0.308667\pi$
$48$	0	0
$49$	−5.30969	−0.758528
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.63508	−1.18612	−0.593060	−	0.805159i	$-0.702080\pi$
$53$	−8.63508	−1.18612	−0.593060	+	0.805159i	$0.702080\pi$
$54$	0	0
$55$	−3.20642	−0.432354
$56$	0	0
$57$	0.0523478	0.00693364
$58$	0	0
$59$	7.72763	1.00605	0.503026	−	0.864271i	$-0.332220\pi$
$59$	7.72763	1.00605	0.503026	+	0.864271i	$0.332220\pi$
$60$	0	0
$61$	6.99620	0.895771	0.447886	−	0.894091i	$-0.352177\pi$
$61$	6.99620	0.895771	0.447886	+	0.894091i	$0.352177\pi$
$62$	0	0
$63$	−1.30012	−0.163799
$64$	0	0
$65$	−20.8104	−2.58121
$66$	0	0
$67$	0.660863	0.0807373	0.0403687	−	0.999185i	$-0.487147\pi$
$67$	0.660863	0.0807373	0.0403687	+	0.999185i	$0.487147\pi$
$68$	0	0
$69$	−2.44426	−0.294254
$70$	0	0
$71$	−7.45066	−0.884231	−0.442115	−	0.896958i	$-0.645772\pi$
$71$	−7.45066	−0.884231	−0.442115	+	0.896958i	$0.645772\pi$
$72$	0	0
$73$	1.38604	0.162224	0.0811118	−	0.996705i	$-0.474153\pi$
$73$	1.38604	0.162224	0.0811118	+	0.996705i	$0.474153\pi$
$74$	0	0
$75$	−4.69014	−0.541571
$76$	0	0
$77$	1.33918	0.152614
$78$	0	0
$79$	−16.6708	−1.87561	−0.937804	−	0.347166i	$-0.887144\pi$
$79$	−16.6708	−1.87561	−0.937804	+	0.347166i	$0.887144\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.05834	0.774753	0.387377	−	0.921922i	$-0.373381\pi$
$83$	7.05834	0.774753	0.387377	+	0.921922i	$0.373381\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.16182	0.124560
$88$	0	0
$89$	−18.5036	−1.96138	−0.980688	−	0.195579i	$-0.937341\pi$
$89$	−18.5036	−1.96138	−0.980688	+	0.195579i	$0.937341\pi$
$90$	0	0
$91$	8.69155	0.911122
$92$	0	0
$93$	−7.49914	−0.777625
$94$	0	0
$95$	−0.162953	−0.0167187
$96$	0	0
$97$	−7.51406	−0.762937	−0.381469	−	0.924382i	$-0.624582\pi$
$97$	−7.51406	−0.762937	−0.381469	+	0.924382i	$0.624582\pi$
$98$	0	0
$99$	−1.03004	−0.103523
$100$	0	0
$101$	−9.34462	−0.929825	−0.464912	−	0.885357i	$-0.653914\pi$
$101$	−9.34462	−0.929825	−0.464912	+	0.885357i	$0.653914\pi$
$102$	0	0
$103$	−5.52226	−0.544125	−0.272062	−	0.962280i	$-0.587706\pi$
$103$	−5.52226	−0.544125	−0.272062	+	0.962280i	$0.587706\pi$
$104$	0	0
$105$	4.04713	0.394960
$106$	0	0
$107$	−5.73905	−0.554815	−0.277407	−	0.960752i	$-0.589475\pi$
$107$	−5.73905	−0.554815	−0.277407	+	0.960752i	$0.589475\pi$
$108$	0	0
$109$	3.40891	0.326514	0.163257	−	0.986584i	$-0.447800\pi$
$109$	3.40891	0.326514	0.163257	+	0.986584i	$0.447800\pi$
$110$	0	0
$111$	4.26691	0.404997
$112$	0	0
$113$	−14.5959	−1.37307	−0.686533	−	0.727098i	$-0.740868\pi$
$113$	−14.5959	−1.37307	−0.686533	+	0.727098i	$0.740868\pi$
$114$	0	0
$115$	7.60873	0.709517
$116$	0	0
$117$	−6.68521	−0.618047
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.93901	−0.903546
$122$	0	0
$123$	2.59407	0.233900
$124$	0	0
$125$	−0.964568	−0.0862736
$126$	0	0
$127$	19.1923	1.70304	0.851520	−	0.524322i	$-0.175681\pi$
$127$	19.1923	1.70304	0.851520	+	0.524322i	$0.175681\pi$
$128$	0	0
$129$	−11.6733	−1.02778
$130$	0	0
$131$	−19.3293	−1.68881	−0.844406	−	0.535703i	$-0.820047\pi$
$131$	−19.3293	−1.68881	−0.844406	+	0.535703i	$0.820047\pi$
$132$	0	0
$133$	0.0680583	0.00590140
$134$	0	0
$135$	−3.11290	−0.267916
$136$	0	0
$137$	−0.386209	−0.0329961	−0.0164980	−	0.999864i	$-0.505252\pi$
$137$	−0.386209	−0.0329961	−0.0164980	+	0.999864i	$0.505252\pi$
$138$	0	0
$139$	−14.6317	−1.24104	−0.620521	−	0.784190i	$-0.713079\pi$
$139$	−14.6317	−1.24104	−0.620521	+	0.784190i	$0.713079\pi$
$140$	0	0
$141$	−7.75431	−0.653031
$142$	0	0
$143$	6.88606	0.575841
$144$	0	0
$145$	−3.61662	−0.300344
$146$	0	0
$147$	5.30969	0.437936
$148$	0	0
$149$	14.8965	1.22037	0.610184	−	0.792260i	$-0.291095\pi$
$149$	14.8965	1.22037	0.610184	+	0.792260i	$0.291095\pi$
$150$	0	0
$151$	−16.4791	−1.34105	−0.670525	−	0.741887i	$-0.733931\pi$
$151$	−16.4791	−1.34105	−0.670525	+	0.741887i	$0.733931\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	23.3441	1.87504
$156$	0	0
$157$	−2.21102	−0.176459	−0.0882294	−	0.996100i	$-0.528121\pi$
$157$	−2.21102	−0.176459	−0.0882294	+	0.996100i	$0.528121\pi$
$158$	0	0
$159$	8.63508	0.684806
$160$	0	0
$161$	−3.17782	−0.250447
$162$	0	0
$163$	1.15104	0.0901561	0.0450780	−	0.998983i	$-0.485646\pi$
$163$	1.15104	0.0901561	0.0450780	+	0.998983i	$0.485646\pi$
$164$	0	0
$165$	3.20642	0.249620
$166$	0	0
$167$	−14.3085	−1.10722	−0.553612	−	0.832775i	$-0.686751\pi$
$167$	−14.3085	−1.10722	−0.553612	+	0.832775i	$0.686751\pi$
$168$	0	0
$169$	31.6920	2.43784
$170$	0	0
$171$	−0.0523478	−0.00400314
$172$	0	0
$173$	−12.2568	−0.931867	−0.465933	−	0.884820i	$-0.654281\pi$
$173$	−12.2568	−0.931867	−0.465933	+	0.884820i	$0.654281\pi$
$174$	0	0
$175$	−6.09773	−0.460945
$176$	0	0
$177$	−7.72763	−0.580844
$178$	0	0
$179$	−18.9999	−1.42012	−0.710061	−	0.704141i	$-0.751332\pi$
$179$	−18.9999	−1.42012	−0.710061	+	0.704141i	$0.751332\pi$
$180$	0	0
$181$	24.6208	1.83005	0.915024	−	0.403399i	$-0.132171\pi$
$181$	24.6208	1.83005	0.915024	+	0.403399i	$0.132171\pi$
$182$	0	0
$183$	−6.99620	−0.517174
$184$	0	0
$185$	−13.2824	−0.976545
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.30012	0.0945696
$190$	0	0
$191$	−2.98494	−0.215983	−0.107991	−	0.994152i	$-0.534442\pi$
$191$	−2.98494	−0.215983	−0.107991	+	0.994152i	$0.534442\pi$
$192$	0	0
$193$	−8.68314	−0.625026	−0.312513	−	0.949913i	$-0.601171\pi$
$193$	−8.68314	−0.625026	−0.312513	+	0.949913i	$0.601171\pi$
$194$	0	0
$195$	20.8104	1.49026
$196$	0	0
$197$	−0.995496	−0.0709262	−0.0354631	−	0.999371i	$-0.511291\pi$
$197$	−0.995496	−0.0709262	−0.0354631	+	0.999371i	$0.511291\pi$
$198$	0	0
$199$	−16.7979	−1.19077	−0.595386	−	0.803440i	$-0.703001\pi$
$199$	−16.7979	−1.19077	−0.595386	+	0.803440i	$0.703001\pi$
$200$	0	0
$201$	−0.660863	−0.0466137
$202$	0	0
$203$	1.51050	0.106016
$204$	0	0
$205$	−8.07509	−0.563989
$206$	0	0
$207$	2.44426	0.169888
$208$	0	0
$209$	0.0539206	0.00372976
$210$	0	0
$211$	7.91393	0.544817	0.272409	−	0.962182i	$-0.412180\pi$
$211$	7.91393	0.544817	0.272409	+	0.962182i	$0.412180\pi$
$212$	0	0
$213$	7.45066	0.510511
$214$	0	0
$215$	36.3378	2.47822
$216$	0	0
$217$	−9.74977	−0.661857
$218$	0	0
$219$	−1.38604	−0.0936598
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−13.8645	−0.928438	−0.464219	−	0.885720i	$-0.653665\pi$
$223$	−13.8645	−0.928438	−0.464219	+	0.885720i	$0.653665\pi$
$224$	0	0
$225$	4.69014	0.312676
$226$	0	0
$227$	4.59293	0.304843	0.152422	−	0.988316i	$-0.451293\pi$
$227$	4.59293	0.304843	0.152422	+	0.988316i	$0.451293\pi$
$228$	0	0
$229$	−15.9245	−1.05232	−0.526160	−	0.850385i	$-0.676369\pi$
$229$	−15.9245	−1.05232	−0.526160	+	0.850385i	$0.676369\pi$
$230$	0	0
$231$	−1.33918	−0.0881115
$232$	0	0
$233$	−19.8586	−1.30098	−0.650491	−	0.759514i	$-0.725437\pi$
$233$	−19.8586	−1.30098	−0.650491	+	0.759514i	$0.725437\pi$
$234$	0	0
$235$	24.1384	1.57461
$236$	0	0
$237$	16.6708	1.08288
$238$	0	0
$239$	5.45935	0.353136	0.176568	−	0.984288i	$-0.443500\pi$
$239$	5.45935	0.353136	0.176568	+	0.984288i	$0.443500\pi$
$240$	0	0
$241$	−2.54493	−0.163933	−0.0819667	−	0.996635i	$-0.526120\pi$
$241$	−2.54493	−0.163933	−0.0819667	+	0.996635i	$0.526120\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−16.5285	−1.05597
$246$	0	0
$247$	0.349956	0.0222672
$248$	0	0
$249$	−7.05834	−0.447304
$250$	0	0
$251$	−5.68072	−0.358564	−0.179282	−	0.983798i	$-0.557377\pi$
$251$	−5.68072	−0.358564	−0.179282	+	0.983798i	$0.557377\pi$
$252$	0	0
$253$	−2.51769	−0.158286
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.44168	0.526578	0.263289	−	0.964717i	$-0.415193\pi$
$257$	8.44168	0.526578	0.263289	+	0.964717i	$0.415193\pi$
$258$	0	0
$259$	5.54748	0.344704
$260$	0	0
$261$	−1.16182	−0.0719148
$262$	0	0
$263$	24.2537	1.49555	0.747775	−	0.663953i	$-0.231122\pi$
$263$	24.2537	1.49555	0.747775	+	0.663953i	$0.231122\pi$
$264$	0	0
$265$	−26.8801	−1.65123
$266$	0	0
$267$	18.5036	1.13240
$268$	0	0
$269$	−5.14984	−0.313992	−0.156996	−	0.987599i	$-0.550181\pi$
$269$	−5.14984	−0.313992	−0.156996	+	0.987599i	$0.550181\pi$
$270$	0	0
$271$	15.6193	0.948808	0.474404	−	0.880307i	$-0.342664\pi$
$271$	15.6193	0.948808	0.474404	+	0.880307i	$0.342664\pi$
$272$	0	0
$273$	−8.69155	−0.526037
$274$	0	0
$275$	−4.83105	−0.291323
$276$	0	0
$277$	−0.326067	−0.0195914	−0.00979572	−	0.999952i	$-0.503118\pi$
$277$	−0.326067	−0.0195914	−0.00979572	+	0.999952i	$0.503118\pi$
$278$	0	0
$279$	7.49914	0.448962
$280$	0	0
$281$	28.0010	1.67040	0.835200	−	0.549947i	$-0.185352\pi$
$281$	28.0010	1.67040	0.835200	+	0.549947i	$0.185352\pi$
$282$	0	0
$283$	15.7175	0.934308	0.467154	−	0.884176i	$-0.345279\pi$
$283$	15.7175	0.934308	0.467154	+	0.884176i	$0.345279\pi$
$284$	0	0
$285$	0.162953	0.00965253
$286$	0	0
$287$	3.37260	0.199078
$288$	0	0
$289$	0	0
$290$	0	0
$291$	7.51406	0.440482
$292$	0	0
$293$	17.3226	1.01200	0.505998	−	0.862535i	$-0.331124\pi$
$293$	17.3226	1.01200	0.505998	+	0.862535i	$0.331124\pi$
$294$	0	0
$295$	24.0553	1.40056
$296$	0	0
$297$	1.03004	0.0597692
$298$	0	0
$299$	−16.3404	−0.944988
$300$	0	0
$301$	−15.1767	−0.874768
$302$	0	0
$303$	9.34462	0.536834
$304$	0	0
$305$	21.7785	1.24703
$306$	0	0
$307$	−18.5111	−1.05649	−0.528244	−	0.849093i	$-0.677149\pi$
$307$	−18.5111	−1.05649	−0.528244	+	0.849093i	$0.677149\pi$
$308$	0	0
$309$	5.52226	0.314151
$310$	0	0
$311$	−27.1945	−1.54206	−0.771030	−	0.636799i	$-0.780258\pi$
$311$	−27.1945	−1.54206	−0.771030	+	0.636799i	$0.780258\pi$
$312$	0	0
$313$	−3.74485	−0.211671	−0.105836	−	0.994384i	$-0.533752\pi$
$313$	−3.74485	−0.211671	−0.105836	+	0.994384i	$0.533752\pi$
$314$	0	0
$315$	−4.04713	−0.228030
$316$	0	0
$317$	13.2121	0.742067	0.371034	−	0.928619i	$-0.379003\pi$
$317$	13.2121	0.742067	0.371034	+	0.928619i	$0.379003\pi$
$318$	0	0
$319$	1.19672	0.0670037
$320$	0	0
$321$	5.73905	0.320323
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−31.3545	−1.73924
$326$	0	0
$327$	−3.40891	−0.188513
$328$	0	0
$329$	−10.0815	−0.555812
$330$	0	0
$331$	−4.54231	−0.249668	−0.124834	−	0.992178i	$-0.539840\pi$
$331$	−4.54231	−0.249668	−0.124834	+	0.992178i	$0.539840\pi$
$332$	0	0
$333$	−4.26691	−0.233825
$334$	0	0
$335$	2.05720	0.112397
$336$	0	0
$337$	0.958938	0.0522367	0.0261183	−	0.999659i	$-0.491685\pi$
$337$	0.958938	0.0522367	0.0261183	+	0.999659i	$0.491685\pi$
$338$	0	0
$339$	14.5959	0.792741
$340$	0	0
$341$	−7.72445	−0.418302
$342$	0	0
$343$	16.0040	0.864137
$344$	0	0
$345$	−7.60873	−0.409640
$346$	0	0
$347$	6.71261	0.360352	0.180176	−	0.983634i	$-0.442333\pi$
$347$	6.71261	0.360352	0.180176	+	0.983634i	$0.442333\pi$
$348$	0	0
$349$	−36.3697	−1.94683	−0.973413	−	0.229058i	$-0.926435\pi$
$349$	−36.3697	−1.94683	−0.973413	+	0.229058i	$0.926435\pi$
$350$	0	0
$351$	6.68521	0.356830
$352$	0	0
$353$	14.0435	0.747460	0.373730	−	0.927538i	$-0.378079\pi$
$353$	14.0435	0.747460	0.373730	+	0.927538i	$0.378079\pi$
$354$	0	0
$355$	−23.1932	−1.23096
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.2902	1.59866	0.799328	−	0.600896i	$-0.205189\pi$
$359$	30.2902	1.59866	0.799328	+	0.600896i	$0.205189\pi$
$360$	0	0
$361$	−18.9973	−0.999856
$362$	0	0
$363$	9.93901	0.521663
$364$	0	0
$365$	4.31460	0.225836
$366$	0	0
$367$	−20.1514	−1.05190	−0.525948	−	0.850517i	$-0.676289\pi$
$367$	−20.1514	−1.05190	−0.525948	+	0.850517i	$0.676289\pi$
$368$	0	0
$369$	−2.59407	−0.135042
$370$	0	0
$371$	11.2266	0.582857
$372$	0	0
$373$	21.6908	1.12311	0.561553	−	0.827441i	$-0.310204\pi$
$373$	21.6908	1.12311	0.561553	+	0.827441i	$0.310204\pi$
$374$	0	0
$375$	0.964568	0.0498101
$376$	0	0
$377$	7.76700	0.400021
$378$	0	0
$379$	−15.6393	−0.803336	−0.401668	−	0.915785i	$-0.631569\pi$
$379$	−15.6393	−0.803336	−0.401668	+	0.915785i	$0.631569\pi$
$380$	0	0
$381$	−19.1923	−0.983251
$382$	0	0
$383$	9.83021	0.502300	0.251150	−	0.967948i	$-0.419191\pi$
$383$	9.83021	0.502300	0.251150	+	0.967948i	$0.419191\pi$
$384$	0	0
$385$	4.16873	0.212458
$386$	0	0
$387$	11.6733	0.593387
$388$	0	0
$389$	10.4247	0.528553	0.264277	−	0.964447i	$-0.414867\pi$
$389$	10.4247	0.528553	0.264277	+	0.964447i	$0.414867\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	19.3293	0.975036
$394$	0	0
$395$	−51.8944	−2.61109
$396$	0	0
$397$	23.0726	1.15798	0.578991	−	0.815334i	$-0.303447\pi$
$397$	23.0726	1.15798	0.578991	+	0.815334i	$0.303447\pi$
$398$	0	0
$399$	−0.0680583	−0.00340718
$400$	0	0
$401$	−18.2794	−0.912831	−0.456415	−	0.889767i	$-0.650867\pi$
$401$	−18.2794	−0.912831	−0.456415	+	0.889767i	$0.650867\pi$
$402$	0	0
$403$	−50.1333	−2.49732
$404$	0	0
$405$	3.11290	0.154681
$406$	0	0
$407$	4.39510	0.217857
$408$	0	0
$409$	11.9639	0.591574	0.295787	−	0.955254i	$-0.404418\pi$
$409$	11.9639	0.591574	0.295787	+	0.955254i	$0.404418\pi$
$410$	0	0
$411$	0.386209	0.0190503
$412$	0	0
$413$	−10.0468	−0.494372
$414$	0	0
$415$	21.9719	1.07856
$416$	0	0
$417$	14.6317	0.716516
$418$	0	0
$419$	−17.1700	−0.838808	−0.419404	−	0.907800i	$-0.637761\pi$
$419$	−17.1700	−0.838808	−0.419404	+	0.907800i	$0.637761\pi$
$420$	0	0
$421$	−22.0942	−1.07680	−0.538402	−	0.842688i	$-0.680972\pi$
$421$	−22.0942	−1.07680	−0.538402	+	0.842688i	$0.680972\pi$
$422$	0	0
$423$	7.75431	0.377027
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−9.09588	−0.440180
$428$	0	0
$429$	−6.88606	−0.332462
$430$	0	0
$431$	−40.2031	−1.93652	−0.968259	−	0.249951i	$-0.919586\pi$
$431$	−40.2031	−1.93652	−0.968259	+	0.249951i	$0.919586\pi$
$432$	0	0
$433$	−5.64686	−0.271371	−0.135685	−	0.990752i	$-0.543324\pi$
$433$	−5.64686	−0.271371	−0.135685	+	0.990752i	$0.543324\pi$
$434$	0	0
$435$	3.61662	0.173404
$436$	0	0
$437$	−0.127952	−0.00612075
$438$	0	0
$439$	−29.3026	−1.39854	−0.699269	−	0.714859i	$-0.746491\pi$
$439$	−29.3026	−1.39854	−0.699269	+	0.714859i	$0.746491\pi$
$440$	0	0
$441$	−5.30969	−0.252843
$442$	0	0
$443$	−32.1170	−1.52592	−0.762962	−	0.646443i	$-0.776256\pi$
$443$	−32.1170	−1.52592	−0.762962	+	0.646443i	$0.776256\pi$
$444$	0	0
$445$	−57.5998	−2.73049
$446$	0	0
$447$	−14.8965	−0.704580
$448$	0	0
$449$	−33.5547	−1.58354	−0.791772	−	0.610817i	$-0.790841\pi$
$449$	−33.5547	−1.58354	−0.791772	+	0.610817i	$0.790841\pi$
$450$	0	0
$451$	2.67201	0.125820
$452$	0	0
$453$	16.4791	0.774256
$454$	0	0
$455$	27.0559	1.26840
$456$	0	0
$457$	37.8941	1.77261	0.886307	−	0.463099i	$-0.153263\pi$
$457$	37.8941	1.77261	0.886307	+	0.463099i	$0.153263\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5.43265	−0.253024	−0.126512	−	0.991965i	$-0.540378\pi$
$461$	−5.43265	−0.253024	−0.126512	+	0.991965i	$0.540378\pi$
$462$	0	0
$463$	−22.0011	−1.02248	−0.511239	−	0.859439i	$-0.670813\pi$
$463$	−22.0011	−1.02248	−0.511239	+	0.859439i	$0.670813\pi$
$464$	0	0
$465$	−23.3441	−1.08256
$466$	0	0
$467$	10.2604	0.474796	0.237398	−	0.971412i	$-0.423705\pi$
$467$	10.2604	0.474796	0.237398	+	0.971412i	$0.423705\pi$
$468$	0	0
$469$	−0.859200	−0.0396742
$470$	0	0
$471$	2.21102	0.101879
$472$	0	0
$473$	−12.0240	−0.552865
$474$	0	0
$475$	−0.245518	−0.0112652
$476$	0	0
$477$	−8.63508	−0.395373
$478$	0	0
$479$	11.1641	0.510100	0.255050	−	0.966928i	$-0.417908\pi$
$479$	11.1641	0.510100	0.255050	+	0.966928i	$0.417908\pi$
$480$	0	0
$481$	28.5251	1.30063
$482$	0	0
$483$	3.17782	0.144596
$484$	0	0
$485$	−23.3905	−1.06211
$486$	0	0
$487$	−31.0204	−1.40567	−0.702833	−	0.711355i	$-0.748082\pi$
$487$	−31.0204	−1.40567	−0.702833	+	0.711355i	$0.748082\pi$
$488$	0	0
$489$	−1.15104	−0.0520516
$490$	0	0
$491$	−3.40267	−0.153560	−0.0767801	−	0.997048i	$-0.524464\pi$
$491$	−3.40267	−0.153560	−0.0767801	+	0.997048i	$0.524464\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−3.20642	−0.144118
$496$	0	0
$497$	9.68673	0.434509
$498$	0	0
$499$	−37.7201	−1.68858	−0.844292	−	0.535883i	$-0.819979\pi$
$499$	−37.7201	−1.68858	−0.844292	+	0.535883i	$0.819979\pi$
$500$	0	0
$501$	14.3085	0.639256
$502$	0	0
$503$	44.1116	1.96684	0.983419	−	0.181349i	$-0.0580463\pi$
$503$	44.1116	1.96684	0.983419	+	0.181349i	$0.0580463\pi$
$504$	0	0
$505$	−29.0889	−1.29444
$506$	0	0
$507$	−31.6920	−1.40749
$508$	0	0
$509$	19.4799	0.863431	0.431716	−	0.902010i	$-0.357908\pi$
$509$	19.4799	0.863431	0.431716	+	0.902010i	$0.357908\pi$
$510$	0	0
$511$	−1.80201	−0.0797163
$512$	0	0
$513$	0.0523478	0.00231121
$514$	0	0
$515$	−17.1902	−0.757493
$516$	0	0
$517$	−7.98728	−0.351280
$518$	0	0
$519$	12.2568	0.538014
$520$	0	0
$521$	2.19708	0.0962557	0.0481279	−	0.998841i	$-0.484675\pi$
$521$	2.19708	0.0962557	0.0481279	+	0.998841i	$0.484675\pi$
$522$	0	0
$523$	28.7418	1.25679	0.628395	−	0.777895i	$-0.283712\pi$
$523$	28.7418	1.25679	0.628395	+	0.777895i	$0.283712\pi$
$524$	0	0
$525$	6.09773	0.266127
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−17.0256	−0.740244
$530$	0	0
$531$	7.72763	0.335351
$532$	0	0
$533$	17.3419	0.751162
$534$	0	0
$535$	−17.8651	−0.772375
$536$	0	0
$537$	18.9999	0.819907
$538$	0	0
$539$	5.46922	0.235576
$540$	0	0
$541$	12.9282	0.555828	0.277914	−	0.960606i	$-0.410357\pi$
$541$	12.9282	0.555828	0.277914	+	0.960606i	$0.410357\pi$
$542$	0	0
$543$	−24.6208	−1.05658
$544$	0	0
$545$	10.6116	0.454551
$546$	0	0
$547$	−2.09260	−0.0894732	−0.0447366	−	0.998999i	$-0.514245\pi$
$547$	−2.09260	−0.0894732	−0.0447366	+	0.998999i	$0.514245\pi$
$548$	0	0
$549$	6.99620	0.298590
$550$	0	0
$551$	0.0608186	0.00259096
$552$	0	0
$553$	21.6739	0.921670
$554$	0	0
$555$	13.2824	0.563808
$556$	0	0
$557$	−15.7425	−0.667029	−0.333515	−	0.942745i	$-0.608235\pi$
$557$	−15.7425	−0.667029	−0.333515	+	0.942745i	$0.608235\pi$
$558$	0	0
$559$	−78.0384	−3.30067
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9.70565	−0.409044	−0.204522	−	0.978862i	$-0.565564\pi$
$563$	−9.70565	−0.409044	−0.204522	+	0.978862i	$0.565564\pi$
$564$	0	0
$565$	−45.4356	−1.91149
$566$	0	0
$567$	−1.30012	−0.0545998
$568$	0	0
$569$	5.01857	0.210389	0.105195	−	0.994452i	$-0.466453\pi$
$569$	5.01857	0.210389	0.105195	+	0.994452i	$0.466453\pi$
$570$	0	0
$571$	22.6671	0.948587	0.474293	−	0.880367i	$-0.342704\pi$
$571$	22.6671	0.948587	0.474293	+	0.880367i	$0.342704\pi$
$572$	0	0
$573$	2.98494	0.124698
$574$	0	0
$575$	11.4639	0.478078
$576$	0	0
$577$	−17.0022	−0.707810	−0.353905	−	0.935281i	$-0.615146\pi$
$577$	−17.0022	−0.707810	−0.353905	+	0.935281i	$0.615146\pi$
$578$	0	0
$579$	8.68314	0.360859
$580$	0	0
$581$	−9.17667	−0.380712
$582$	0	0
$583$	8.89451	0.368373
$584$	0	0
$585$	−20.8104	−0.860403
$586$	0	0
$587$	16.0672	0.663165	0.331583	−	0.943426i	$-0.392417\pi$
$587$	16.0672	0.663165	0.331583	+	0.943426i	$0.392417\pi$
$588$	0	0
$589$	−0.392564	−0.0161753
$590$	0	0
$591$	0.995496	0.0409493
$592$	0	0
$593$	25.5090	1.04753	0.523764	−	0.851863i	$-0.324527\pi$
$593$	25.5090	1.04753	0.523764	+	0.851863i	$0.324527\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.7979	0.687493
$598$	0	0
$599$	2.88098	0.117714	0.0588568	−	0.998266i	$-0.481254\pi$
$599$	2.88098	0.117714	0.0588568	+	0.998266i	$0.481254\pi$
$600$	0	0
$601$	−26.3975	−1.07678	−0.538388	−	0.842697i	$-0.680966\pi$
$601$	−26.3975	−1.07678	−0.538388	+	0.842697i	$0.680966\pi$
$602$	0	0
$603$	0.660863	0.0269124
$604$	0	0
$605$	−30.9391	−1.25785
$606$	0	0
$607$	−18.7248	−0.760016	−0.380008	−	0.924983i	$-0.624079\pi$
$607$	−18.7248	−0.760016	−0.380008	+	0.924983i	$0.624079\pi$
$608$	0	0
$609$	−1.51050	−0.0612086
$610$	0	0
$611$	−51.8392	−2.09719
$612$	0	0
$613$	−2.20627	−0.0891105	−0.0445552	−	0.999007i	$-0.514187\pi$
$613$	−2.20627	−0.0891105	−0.0445552	+	0.999007i	$0.514187\pi$
$614$	0	0
$615$	8.07509	0.325619
$616$	0	0
$617$	30.0940	1.21154	0.605769	−	0.795641i	$-0.292866\pi$
$617$	30.0940	1.21154	0.605769	+	0.795641i	$0.292866\pi$
$618$	0	0
$619$	11.4620	0.460697	0.230348	−	0.973108i	$-0.426013\pi$
$619$	11.4620	0.460697	0.230348	+	0.973108i	$0.426013\pi$
$620$	0	0
$621$	−2.44426	−0.0980847
$622$	0	0
$623$	24.0568	0.963816
$624$	0	0
$625$	−26.4533	−1.05813
$626$	0	0
$627$	−0.0539206	−0.00215338
$628$	0	0
$629$	0	0
$630$	0	0
$631$	36.5501	1.45503	0.727517	−	0.686089i	$-0.240674\pi$
$631$	36.5501	1.45503	0.727517	+	0.686089i	$0.240674\pi$
$632$	0	0
$633$	−7.91393	−0.314550
$634$	0	0
$635$	59.7436	2.37085
$636$	0	0
$637$	35.4964	1.40642
$638$	0	0
$639$	−7.45066	−0.294744
$640$	0	0
$641$	−15.1654	−0.598997	−0.299499	−	0.954097i	$-0.596819\pi$
$641$	−15.1654	−0.598997	−0.299499	+	0.954097i	$0.596819\pi$
$642$	0	0
$643$	44.1598	1.74149	0.870747	−	0.491732i	$-0.163636\pi$
$643$	44.1598	1.74149	0.870747	+	0.491732i	$0.163636\pi$
$644$	0	0
$645$	−36.3378	−1.43080
$646$	0	0
$647$	−40.4822	−1.59152	−0.795759	−	0.605613i	$-0.792928\pi$
$647$	−40.4822	−1.59152	−0.795759	+	0.605613i	$0.792928\pi$
$648$	0	0
$649$	−7.95980	−0.312450
$650$	0	0
$651$	9.74977	0.382123
$652$	0	0
$653$	−9.19116	−0.359678	−0.179839	−	0.983696i	$-0.557558\pi$
$653$	−9.19116	−0.359678	−0.179839	+	0.983696i	$0.557558\pi$
$654$	0	0
$655$	−60.1703	−2.35105
$656$	0	0
$657$	1.38604	0.0540745
$658$	0	0
$659$	−41.6886	−1.62396	−0.811979	−	0.583686i	$-0.801610\pi$
$659$	−41.6886	−1.62396	−0.811979	+	0.583686i	$0.801610\pi$
$660$	0	0
$661$	35.9912	1.39990	0.699948	−	0.714193i	$-0.253206\pi$
$661$	35.9912	1.39990	0.699948	+	0.714193i	$0.253206\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.211859	0.00821552
$666$	0	0
$667$	−2.83978	−0.109957
$668$	0	0
$669$	13.8645	0.536034
$670$	0	0
$671$	−7.20639	−0.278200
$672$	0	0
$673$	36.6284	1.41192	0.705961	−	0.708250i	$-0.250515\pi$
$673$	36.6284	1.41192	0.705961	+	0.708250i	$0.250515\pi$
$674$	0	0
$675$	−4.69014	−0.180524
$676$	0	0
$677$	−21.5555	−0.828444	−0.414222	−	0.910176i	$-0.635946\pi$
$677$	−21.5555	−0.828444	−0.414222	+	0.910176i	$0.635946\pi$
$678$	0	0
$679$	9.76916	0.374906
$680$	0	0
$681$	−4.59293	−0.176001
$682$	0	0
$683$	36.2097	1.38553	0.692763	−	0.721166i	$-0.256393\pi$
$683$	36.2097	1.38553	0.692763	+	0.721166i	$0.256393\pi$
$684$	0	0
$685$	−1.20223	−0.0459348
$686$	0	0
$687$	15.9245	0.607558
$688$	0	0
$689$	57.7273	2.19923
$690$	0	0
$691$	−2.75554	−0.104826	−0.0524128	−	0.998626i	$-0.516691\pi$
$691$	−2.75554	−0.104826	−0.0524128	+	0.998626i	$0.516691\pi$
$692$	0	0
$693$	1.33918	0.0508712
$694$	0	0
$695$	−45.5469	−1.72769
$696$	0	0
$697$	0	0
$698$	0	0
$699$	19.8586	0.751122
$700$	0	0
$701$	8.81001	0.332750	0.166375	−	0.986063i	$-0.446794\pi$
$701$	8.81001	0.332750	0.166375	+	0.986063i	$0.446794\pi$
$702$	0	0
$703$	0.223363	0.00842430
$704$	0	0
$705$	−24.1384	−0.909104
$706$	0	0
$707$	12.1491	0.456914
$708$	0	0
$709$	40.4085	1.51757	0.758787	−	0.651338i	$-0.225792\pi$
$709$	40.4085	1.51757	0.758787	+	0.651338i	$0.225792\pi$
$710$	0	0
$711$	−16.6708	−0.625202
$712$	0	0
$713$	18.3298	0.686458
$714$	0	0
$715$	21.4356	0.801646
$716$	0	0
$717$	−5.45935	−0.203883
$718$	0	0
$719$	−15.6549	−0.583828	−0.291914	−	0.956445i	$-0.594292\pi$
$719$	−15.6549	−0.583828	−0.291914	+	0.956445i	$0.594292\pi$
$720$	0	0
$721$	7.17959	0.267382
$722$	0	0
$723$	2.54493	0.0946470
$724$	0	0
$725$	−5.44909	−0.202374
$726$	0	0
$727$	−28.5009	−1.05704	−0.528521	−	0.848920i	$-0.677253\pi$
$727$	−28.5009	−1.05704	−0.528521	+	0.848920i	$0.677253\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−17.4730	−0.645379	−0.322690	−	0.946505i	$-0.604587\pi$
$733$	−17.4730	−0.645379	−0.322690	+	0.946505i	$0.604587\pi$
$734$	0	0
$735$	16.5285	0.609664
$736$	0	0
$737$	−0.680719	−0.0250746
$738$	0	0
$739$	42.1606	1.55090	0.775451	−	0.631408i	$-0.217523\pi$
$739$	42.1606	1.55090	0.775451	+	0.631408i	$0.217523\pi$
$740$	0	0
$741$	−0.349956	−0.0128559
$742$	0	0
$743$	25.6779	0.942029	0.471014	−	0.882125i	$-0.343888\pi$
$743$	25.6779	0.942029	0.471014	+	0.882125i	$0.343888\pi$
$744$	0	0
$745$	46.3713	1.69891
$746$	0	0
$747$	7.05834	0.258251
$748$	0	0
$749$	7.46144	0.272635
$750$	0	0
$751$	−23.3913	−0.853562	−0.426781	−	0.904355i	$-0.640352\pi$
$751$	−23.3913	−0.853562	−0.426781	+	0.904355i	$0.640352\pi$
$752$	0	0
$753$	5.68072	0.207017
$754$	0	0
$755$	−51.2978	−1.86692
$756$	0	0
$757$	−36.2123	−1.31616	−0.658079	−	0.752949i	$-0.728631\pi$
$757$	−36.2123	−1.31616	−0.658079	+	0.752949i	$0.728631\pi$
$758$	0	0
$759$	2.51769	0.0913865
$760$	0	0
$761$	6.19717	0.224647	0.112324	−	0.993672i	$-0.464171\pi$
$761$	6.19717	0.224647	0.112324	+	0.993672i	$0.464171\pi$
$762$	0	0
$763$	−4.43198	−0.160449
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−51.6608	−1.86536
$768$	0	0
$769$	−28.8712	−1.04112	−0.520561	−	0.853825i	$-0.674277\pi$
$769$	−28.8712	−1.04112	−0.520561	+	0.853825i	$0.674277\pi$
$770$	0	0
$771$	−8.44168	−0.304020
$772$	0	0
$773$	−6.52574	−0.234715	−0.117357	−	0.993090i	$-0.537442\pi$
$773$	−6.52574	−0.234715	−0.117357	+	0.993090i	$0.537442\pi$
$774$	0	0
$775$	35.1720	1.26342
$776$	0	0
$777$	−5.54748	−0.199015
$778$	0	0
$779$	0.135794	0.00486533
$780$	0	0
$781$	7.67451	0.274616
$782$	0	0
$783$	1.16182	0.0415200
$784$	0	0
$785$	−6.88269	−0.245654
$786$	0	0
$787$	9.04019	0.322248	0.161124	−	0.986934i	$-0.448488\pi$
$787$	9.04019	0.322248	0.161124	+	0.986934i	$0.448488\pi$
$788$	0	0
$789$	−24.2537	−0.863456
$790$	0	0
$791$	18.9764	0.674723
$792$	0	0
$793$	−46.7710	−1.66089
$794$	0	0
$795$	26.8801	0.953340
$796$	0	0
$797$	−4.77716	−0.169216	−0.0846078	−	0.996414i	$-0.526964\pi$
$797$	−4.77716	−0.169216	−0.0846078	+	0.996414i	$0.526964\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−18.5036	−0.653792
$802$	0	0
$803$	−1.42768	−0.0503818
$804$	0	0
$805$	−9.89224	−0.348656
$806$	0	0
$807$	5.14984	0.181283
$808$	0	0
$809$	−30.4539	−1.07070	−0.535351	−	0.844630i	$-0.679820\pi$
$809$	−30.4539	−1.07070	−0.535351	+	0.844630i	$0.679820\pi$
$810$	0	0
$811$	9.45166	0.331893	0.165946	−	0.986135i	$-0.446932\pi$
$811$	9.45166	0.331893	0.165946	+	0.986135i	$0.446932\pi$
$812$	0	0
$813$	−15.6193	−0.547794
$814$	0	0
$815$	3.58306	0.125509
$816$	0	0
$817$	−0.611071	−0.0213787
$818$	0	0
$819$	8.69155	0.303707
$820$	0	0
$821$	25.3144	0.883478	0.441739	−	0.897144i	$-0.354362\pi$
$821$	25.3144	0.883478	0.441739	+	0.897144i	$0.354362\pi$
$822$	0	0
$823$	6.49912	0.226545	0.113273	−	0.993564i	$-0.463867\pi$
$823$	6.49912	0.226545	0.113273	+	0.993564i	$0.463867\pi$
$824$	0	0
$825$	4.83105	0.168196
$826$	0	0
$827$	38.2412	1.32978	0.664889	−	0.746942i	$-0.268479\pi$
$827$	38.2412	1.32978	0.664889	+	0.746942i	$0.268479\pi$
$828$	0	0
$829$	3.18171	0.110505	0.0552526	−	0.998472i	$-0.482404\pi$
$829$	3.18171	0.110505	0.0552526	+	0.998472i	$0.482404\pi$
$830$	0	0
$831$	0.326067	0.0113111
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−44.5409	−1.54140
$836$	0	0
$837$	−7.49914	−0.259208
$838$	0	0
$839$	13.6168	0.470104	0.235052	−	0.971983i	$-0.424474\pi$
$839$	13.6168	0.470104	0.235052	+	0.971983i	$0.424474\pi$
$840$	0	0
$841$	−27.6502	−0.953454
$842$	0	0
$843$	−28.0010	−0.964406
$844$	0	0
$845$	98.6539	3.39380
$846$	0	0
$847$	12.9219	0.444001
$848$	0	0
$849$	−15.7175	−0.539423
$850$	0	0
$851$	−10.4294	−0.357516
$852$	0	0
$853$	−53.9760	−1.84810	−0.924051	−	0.382270i	$-0.875143\pi$
$853$	−53.9760	−1.84810	−0.924051	+	0.382270i	$0.875143\pi$
$854$	0	0
$855$	−0.162953	−0.00557289
$856$	0	0
$857$	27.4344	0.937142	0.468571	−	0.883426i	$-0.344769\pi$
$857$	27.4344	0.937142	0.468571	+	0.883426i	$0.344769\pi$
$858$	0	0
$859$	32.0688	1.09417	0.547087	−	0.837076i	$-0.315737\pi$
$859$	32.0688	1.09417	0.547087	+	0.837076i	$0.315737\pi$
$860$	0	0
$861$	−3.37260	−0.114938
$862$	0	0
$863$	5.64442	0.192138	0.0960692	−	0.995375i	$-0.469373\pi$
$863$	5.64442	0.192138	0.0960692	+	0.995375i	$0.469373\pi$
$864$	0	0
$865$	−38.1542	−1.29728
$866$	0	0
$867$	0	0
$868$	0	0
$869$	17.1716	0.582507
$870$	0	0
$871$	−4.41801	−0.149698
$872$	0	0
$873$	−7.51406	−0.254312
$874$	0	0
$875$	1.25405	0.0423947
$876$	0	0
$877$	−16.6957	−0.563775	−0.281887	−	0.959447i	$-0.590960\pi$
$877$	−16.6957	−0.563775	−0.281887	+	0.959447i	$0.590960\pi$
$878$	0	0
$879$	−17.3226	−0.584276
$880$	0	0
$881$	46.2525	1.55829	0.779144	−	0.626845i	$-0.215654\pi$
$881$	46.2525	1.55829	0.779144	+	0.626845i	$0.215654\pi$
$882$	0	0
$883$	−16.0530	−0.540227	−0.270114	−	0.962828i	$-0.587061\pi$
$883$	−16.0530	−0.540227	−0.270114	+	0.962828i	$0.587061\pi$
$884$	0	0
$885$	−24.0553	−0.808611
$886$	0	0
$887$	18.5689	0.623483	0.311742	−	0.950167i	$-0.399088\pi$
$887$	18.5689	0.623483	0.311742	+	0.950167i	$0.399088\pi$
$888$	0	0
$889$	−24.9522	−0.836871
$890$	0	0
$891$	−1.03004	−0.0345078
$892$	0	0
$893$	−0.405921	−0.0135836
$894$	0	0
$895$	−59.1449	−1.97699
$896$	0	0
$897$	16.3404	0.545589
$898$	0	0
$899$	−8.71264	−0.290583
$900$	0	0
$901$	0	0
$902$	0	0
$903$	15.1767	0.505047
$904$	0	0
$905$	76.6420	2.54767
$906$	0	0
$907$	29.6844	0.985653	0.492826	−	0.870128i	$-0.335964\pi$
$907$	29.6844	0.985653	0.492826	+	0.870128i	$0.335964\pi$
$908$	0	0
$909$	−9.34462	−0.309942
$910$	0	0
$911$	−2.97582	−0.0985933	−0.0492966	−	0.998784i	$-0.515698\pi$
$911$	−2.97582	−0.0985933	−0.0492966	+	0.998784i	$0.515698\pi$
$912$	0	0
$913$	−7.27040	−0.240615
$914$	0	0
$915$	−21.7785	−0.719973
$916$	0	0
$917$	25.1304	0.829879
$918$	0	0
$919$	39.7923	1.31263	0.656313	−	0.754489i	$-0.272115\pi$
$919$	39.7923	1.31263	0.656313	+	0.754489i	$0.272115\pi$
$920$	0	0
$921$	18.5111	0.609963
$922$	0	0
$923$	49.8092	1.63949
$924$	0	0
$925$	−20.0124	−0.658003
$926$	0	0
$927$	−5.52226	−0.181375
$928$	0	0
$929$	−41.2397	−1.35303	−0.676516	−	0.736428i	$-0.736511\pi$
$929$	−41.2397	−1.35303	−0.676516	+	0.736428i	$0.736511\pi$
$930$	0	0
$931$	0.277951	0.00910947
$932$	0	0
$933$	27.1945	0.890309
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−21.6170	−0.706198	−0.353099	−	0.935586i	$-0.614872\pi$
$937$	−21.6170	−0.706198	−0.353099	+	0.935586i	$0.614872\pi$
$938$	0	0
$939$	3.74485	0.122208
$940$	0	0
$941$	32.9254	1.07334	0.536669	−	0.843793i	$-0.319683\pi$
$941$	32.9254	1.07334	0.536669	+	0.843793i	$0.319683\pi$
$942$	0	0
$943$	−6.34059	−0.206478
$944$	0	0
$945$	4.04713	0.131653
$946$	0	0
$947$	28.7813	0.935266	0.467633	−	0.883923i	$-0.345107\pi$
$947$	28.7813	0.935266	0.467633	+	0.883923i	$0.345107\pi$
$948$	0	0
$949$	−9.26595	−0.300785
$950$	0	0
$951$	−13.2121	−0.428433
$952$	0	0
$953$	−49.6066	−1.60692	−0.803458	−	0.595361i	$-0.797009\pi$
$953$	−49.6066	−1.60692	−0.803458	+	0.595361i	$0.797009\pi$
$954$	0	0
$955$	−9.29183	−0.300676
$956$	0	0
$957$	−1.19672	−0.0386846
$958$	0	0
$959$	0.502117	0.0162142
$960$	0	0
$961$	25.2372	0.814102
$962$	0	0
$963$	−5.73905	−0.184938
$964$	0	0
$965$	−27.0297	−0.870118
$966$	0	0
$967$	33.1191	1.06504	0.532520	−	0.846418i	$-0.321245\pi$
$967$	33.1191	1.06504	0.532520	+	0.846418i	$0.321245\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	61.2165	1.96453	0.982265	−	0.187499i	$-0.0600382\pi$
$971$	61.2165	1.96453	0.982265	+	0.187499i	$0.0600382\pi$
$972$	0	0
$973$	19.0229	0.609846
$974$	0	0
$975$	31.3545	1.00415
$976$	0	0
$977$	−7.81686	−0.250083	−0.125042	−	0.992151i	$-0.539906\pi$
$977$	−7.81686	−0.250083	−0.125042	+	0.992151i	$0.539906\pi$
$978$	0	0
$979$	19.0595	0.609145
$980$	0	0
$981$	3.40891	0.108838
$982$	0	0
$983$	22.2548	0.709817	0.354909	−	0.934901i	$-0.384512\pi$
$983$	22.2548	0.709817	0.354909	+	0.934901i	$0.384512\pi$
$984$	0	0
$985$	−3.09888	−0.0987385
$986$	0	0
$987$	10.0815	0.320898
$988$	0	0
$989$	28.5325	0.907282
$990$	0	0
$991$	−16.7755	−0.532891	−0.266445	−	0.963850i	$-0.585849\pi$
$991$	−16.7755	−0.532891	−0.266445	+	0.963850i	$0.585849\pi$
$992$	0	0
$993$	4.54231	0.144146
$994$	0	0
$995$	−52.2902	−1.65771
$996$	0	0
$997$	−42.1065	−1.33353	−0.666763	−	0.745270i	$-0.732321\pi$
$997$	−42.1065	−1.33353	−0.666763	+	0.745270i	$0.732321\pi$
$998$	0	0
$999$	4.26691	0.134999

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6936.2.a.bl.1.8		8
17.11	odd	16		408.2.ba.a.121.4	✓	16
17.14	odd	16		408.2.ba.a.145.4	yes	16
17.16	even	2		6936.2.a.bo.1.1		8
51.11	even	16		1224.2.bq.e.937.1		16
51.14	even	16		1224.2.bq.e.145.1		16
68.11	even	16		816.2.bq.f.529.2		16
68.31	even	16		816.2.bq.f.145.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
408.2.ba.a.121.4	✓	16	17.11	odd	16
408.2.ba.a.145.4	yes	16	17.14	odd	16
816.2.bq.f.145.2		16	68.31	even	16
816.2.bq.f.529.2		16	68.11	even	16
1224.2.bq.e.145.1		16	51.14	even	16
1224.2.bq.e.937.1		16	51.11	even	16
6936.2.a.bl.1.8		8	1.1	even	1	trivial
6936.2.a.bo.1.1		8	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6936 = 2^{3} \cdot 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6936.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.11290 q^{5} -1.30012 q^{7} +1.00000 q^{9} -1.03004 q^{11} -6.68521 q^{13} -3.11290 q^{15} -0.0523478 q^{19} +1.30012 q^{21} +2.44426 q^{23} +4.69014 q^{25} -1.00000 q^{27} -1.16182 q^{29} +7.49914 q^{31} +1.03004 q^{33} -4.04713 q^{35} -4.26691 q^{37} +6.68521 q^{39} -2.59407 q^{41} +11.6733 q^{43} +3.11290 q^{45} +7.75431 q^{47} -5.30969 q^{49} -8.63508 q^{53} -3.20642 q^{55} +0.0523478 q^{57} +7.72763 q^{59} +6.99620 q^{61} -1.30012 q^{63} -20.8104 q^{65} +0.660863 q^{67} -2.44426 q^{69} -7.45066 q^{71} +1.38604 q^{73} -4.69014 q^{75} +1.33918 q^{77} -16.6708 q^{79} +1.00000 q^{81} +7.05834 q^{83} +1.16182 q^{87} -18.5036 q^{89} +8.69155 q^{91} -7.49914 q^{93} -0.162953 q^{95} -7.51406 q^{97} -1.03004 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} - 4 q^{5} + 8 q^{9} - 4 q^{11} - 4 q^{13} + 4 q^{15} + 4 q^{19} + 4 q^{23} - 4 q^{25} - 8 q^{27} - 8 q^{31} + 4 q^{33} + 8 q^{35} - 16 q^{37} + 4 q^{39} - 20 q^{41} + 28 q^{43} - 4 q^{45}+ \cdots - 4 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 - 4 * q^5 + 8 * q^9 - 4 * q^11 - 4 * q^13 + 4 * q^15 + 4 * q^19 + 4 * q^23 - 4 * q^25 - 8 * q^27 - 8 * q^31 + 4 * q^33 + 8 * q^35 - 16 * q^37 + 4 * q^39 - 20 * q^41 + 28 * q^43 - 4 * q^45 + 24 * q^47 - 8 * q^49 + 8 * q^53 - 4 * q^55 - 4 * q^57 + 16 * q^59 - 24 * q^61 - 28 * q^65 - 4 * q^69 - 40 * q^73 + 4 * q^75 + 16 * q^77 - 8 * q^79 + 8 * q^81 + 32 * q^83 + 16 * q^89 + 8 * q^93 - 4 * q^95 - 48 * q^97 - 4 * q^99

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